The stem should root if it had a couple of small roots on it. If they don't have any roots I like to put them in potting soil in bright shade until they lift their heads again which indicates they have rooted. Then into filtered sunlight just like hardening off seedlings or cuttings. Once they have a strong enough root system you can put them back into the garden or grow as a pot root.

Plant the tuber also as two or four more eyes will form and grow from the original eye.

I had a 4 ft stalk of a dahlia break at the tuber last 2 years in a row. In both years, I jammed the whole stalk down into a container and let it go - and they bloomed nicely both years. When I dug them up they had made small tubers off the stem. I think you have to keep the soil moist because it has a lot of green plant growth to transfer water to, so you wouldn't want to stress the plant by drying it out.